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 Cos

 http://functions.wolfram.com/01.07.21.2443.01

 Input Form

 Integrate[z^n Sin[b z^2 + d z + e] Cos[f z + g]^v, z] == 2^(-2 - v) (((-I) ((-I) b)^(-1 - n) Binomial[v, v/2] (-1 + Mod[v, 2]) (E^(2 I e) Sum[2^(-n + q) (I d)^(n - q) ((-I) (d + 2 b z))^(1 + q) (-((I (d + 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d + 2 b z)^2)/(4 b))], {q, 0, n}] - E^((I d^2)/(2 b)) Sum[2^(-n + q) (I d)^(n - q) ((-I) (d + 2 b z))^ (1 + q) ((I (d + 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d + 2 b z)^2)/(4 b)], {q, 0, n}]))/ E^((1/4) I (d^2/b + 4 e)) + I Sum[(Binomial[v, s] ((I b)^(-1 - n) E^(-2 I g (-2 s + v) - (I (-d - 2 f s + f v)^2)/(4 b)) Sum[2^(-n + q) (I (-d - 2 f s + f v))^(n - q) ((-I) (-d + f (-2 s + v) - 2 b z))^(1 + q) (-((I (-d + f (-2 s + v) - 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (-d + f (-2 s + v) - 2 b z)^ 2)/(4 b))], {q, 0, n}] + ((-I) b)^(-1 - n) E^(I (-2 e + Pi + (-d - 2 f s + f v)^2/(4 b))) Sum[2^(-n + q) ((-I) (-d - 2 f s + f v))^(n - q) (I (-d + f (-2 s + v) - 2 b z))^(1 + q) ((I (-d + f (-2 s + v) - 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (-d + f (-2 s + v) - 2 b z)^2)/ (4 b)], {q, 0, n}] + ((I b)^(-1 - n) Sum[2^(-n + q) (I (-d + 2 f s - f v))^(n - q) ((-I) (-d + 2 f s - f v - 2 b z))^(1 + q) (-((I (d + f (-2 s + v) + 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d + f (-2 s + v) + 2 b z)^ 2)/(4 b))], {q, 0, n}])/E^((I (d + f (-2 s + v))^2)/ (4 b)) + ((-I) b)^(-1 - n) E^(I (-2 e + Pi - 2 g (-2 s + v) + (d + f (-2 s + v))^2/(4 b))) Sum[2^(-n + q) ((-I) (-d + 2 f s - f v))^(n - q) (I (-d + 2 f s - f v - 2 b z))^ (1 + q) ((I (d + f (-2 s + v) + 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d + f (-2 s + v) + 2 b z)^2)/ (4 b)], {q, 0, n}]))/E^(I (-e + 2 g s - g v)), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin ( b z 2 + d z + e ) cos v ( f z + g ) z 2 - v - 2 ( s = 0 v - 1 2 - ( - e + 2 g s - g v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( d + f ( v - 2 s ) ) 2 4 b - 2 e - 2 g ( v - 2 s ) + π ) ( q = 0 n 2 q - n ( - ( - d + 2 f s - f v ) ) n - q ( ( - d + 2 f s - f v - 2 b z ) ) q + 1 ( ( d + f ( v - 2 s ) + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( d + f ( v - 2 s ) + 2 b z ) 2 4 b ) ) ( - b ) - n - 1 + ( ( - d - 2 f s + f v ) 2 4 b - 2 e + π ) ( q = 0 n 2 q - n ( - ( - d - 2 f s + f v ) ) n - q ( ( - d + f ( v - 2 s ) - 2 b z ) ) q + 1 ( ( - d + f ( v - 2 s ) - 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( - d + f ( v - 2 s ) - 2 b z ) 2 4 b ) ) ( - b ) - n - 1 + ( b ) - n - 1 - ( d + f ( v - 2 s ) ) 2 4 b q = 0 n 2 q - n ( ( - d + 2 f s - f v ) ) n - q ( - ( - d + 2 f s - f v - 2 b z ) ) q + 1 ( - ( d + f ( v - 2 s ) + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( d + f ( v - 2 s ) + 2 b z ) 2 4 b ) + ( b ) - n - 1 - ( - d - 2 f s + f v ) 2 4 b - 2 g ( v - 2 s ) q = 0 n 2 q - n ( ( - d - 2 f s + f v ) ) n - q ( - ( - d + f ( v - 2 s ) - 2 b z ) ) q + 1 ( - ( - d + f ( v - 2 s ) - 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( - d + f ( v - 2 s ) - 2 b z ) 2 4 b ) ) - ( - b ) - n - 1 - 1 4 ( d 2 b + 4 e ) ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) ( 2 e q = 0 n 2 q - n ( d ) n - q ( - ( d + 2 b z ) ) q + 1 ( - ( d + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( d + 2 b z ) 2 4 b ) - d 2 2 b q = 0 n 2 q - n ( d ) n - q ( - ( d + 2 b z ) ) q + 1 ( ( d + 2 b z ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( d + 2 b z ) 2 4 b ) ) ) /; n v + Condition z z n b z 2 d z e f z g v 2 -1 v -2 s 0 v -1 2 -1 -1 -1 e 2 g s -1 g v Binomial v s d f v -1 2 s 2 4 b -1 -1 2 e -1 2 g v -1 2 s q 0 n 2 q -1 n -1 -1 d 2 f s -1 f v n -1 q -1 d 2 f s -1 f v -1 2 b z q 1 d f v -1 2 s 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d f v -1 2 s 2 b z 2 4 b -1 -1 b -1 n -1 -1 d -1 2 f s f v 2 4 b -1 -1 2 e q 0 n 2 q -1 n -1 -1 d -1 2 f s f v n -1 q -1 d f v -1 2 s -1 2 b z q 1 -1 d f v -1 2 s -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d f v -1 2 s -1 2 b z 2 4 b -1 -1 b -1 n -1 b -1 n -1 -1 d f v -1 2 s 2 4 b -1 q 0 n 2 q -1 n -1 d 2 f s -1 f v n -1 q -1 -1 d 2 f s -1 f v -1 2 b z q 1 -1 d f v -1 2 s 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d f v -1 2 s 2 b z 2 4 b -1 b -1 n -1 -1 -1 d -1 2 f s f v 2 4 b -1 -1 2 g v -1 2 s q 0 n 2 q -1 n -1 d -1 2 f s f v n -1 q -1 -1 d f v -1 2 s -1 2 b z q 1 -1 -1 d f v -1 2 s -1 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 d f v -1 2 s -1 2 b z 2 4 b -1 -1 -1 b -1 n -1 -1 1 4 d 2 b -1 4 e Binomial v v 2 -1 \$CellContext`v 2 -1 2 e q 0 n 2 q -1 n d n -1 q -1 d 2 b z q 1 -1 d 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d 2 b z 2 4 b -1 -1 d 2 2 b -1 q 0 n 2 q -1 n d n -1 q -1 d 2 b z q 1 d 2 b z 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d 2 b z 2 4 b -1 n v SuperPlus [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18